On forced dynamical systems with a singularity of repulsive type

rlbnlrnear Analym. Theory, Pruned in Great Bntam.

Merhods

ON FORCED

& A~,,lrcarions.

Vol.

14. No

6. pp. 4X9-500.

19%.

0362-546X.90 53.00+ .@I C 1990 Pergamon Press plc

DYNAMICAL SYSTEMS WITH A SINGULARITY REPULSIVE TYPE

SERGIO SOLIMINI International School for Advanced Studies (SISSA/ISAS), (Received

19 September

1988; received for publication

OF

Trieste. Italy

18 April 1989)

Key words and phrases:

Second order O.D.E. systems, potentials with a singularity near which the gradient is inward directed, periodic solutions.

I. INTRODUCTION IN THIS

paper we discuss the existence of a periodic solution to the forced dynamical system -2 = VF(x)

+ h(f)

(P)

x(t + T) = x(t)

where F is a potential with a singularity in zero of repulsive type like, for instance, the electrostatic potential between two charges of the same sign. More precisely F E C’(lR”\(O), R) and F is asked to satisfy the following condition limF(x)

= +co.

x-0

(P, )

We consider the problem of finding conditions on the T-periodic continuous forcing term h which make (P) have, or have not, a solution and we generalize in this way a result obtained for the scalar case n = 1 in a joint paper with Lazer [5]. Condition (F,) makes one expect the gradient field VF to be inward directed. We formalize this assumption by considering two cases of different generality. There exist constants c, , c2 such that v x E IT?\(O): VF(x)-x I cr + cZlxl li;t;p

VF(x)-fi

(P,)

< 0.

There exists E > 0 such that

(P,)’ < E one has VF(x)-y < 0.

(FZ) is clearly much weaker than (F,)‘. Since we also ask to the gradient field that it vanishes at infinity, namely that lim VF(x) = 0, (PJ ) 1x1-m 489

s. SoLrMINr

490

then condition (F2) simply means that the outward radial component of VF(x) can grow at most like [xl-’ as x -+ 0. This looks to be very natural under (F,) and can be also further weakened by a more general bound which involves the speed of the growth of F. Set S’ = [0, T]/[O, TJ. Our results are as follows. THEOREM1. Assume (F,)’ holds. Then rl > 0 exists such that, for every h E C(S’) such that

lbllrn< tl and

T h(t) dr = 0, 0

(P) has no solution. THEOREM2. Assume (F,)-(F,)-(F,)

hold. Then, for every h E C(S’) such that T h(t) dl

# 0,

s0 (P) has at least one solution. In order to consider cases in which (F,) holds but (F,)’ does not we also state the condition 3R>OsuchthatvyEIR”,(y(2R:F~)IN:

=,,tiy

-402

F(y)

(Fd)

and if IyI = R the strict inequality holds. Assumption (F4) considers a case in which the potential becomes attractive far away from the origin and it is therefore in contradiction to (F2)‘. In fact we have the following. THEOREM3. Assume (F,), (F2), (F3) and (F4) hold. Then for every h E C(S’), (P) has at least one solution. Two particular cases included in (F4) are the case in which F tends to +co at infinity and the case considered by Coti Zelati [2, theorem 21 under some technical restrictions that, in this way, we show, of course, to be unnecessary. In fact the proof does not change if one considers F to be t-depending and then one gets [2, theorem 21 by taking h = 0. In the same paper the problem is considered of avoiding constant solutions, which must necessarily exist, as a consequence of (F,), (F4), in the case in which h = 0. Since our method of proof is based on Rabinowitz’s saddle point theorem, we get an immediate corollary of theorem 3, which ensures the existence of a nonconstant solution to (P), for T large, provided F has only finitely many nondegenerate critical points, as a consequence of the characterization of the Morse index, see [2]. Therefore, [2, theorem I] is also improved by our approach. Finally, this result is in some sense sharp because if we replace (Fz) by a stronger condition (F,)“, weaker than (F,)’ in such a way to be still compatible with (F4), nonconstant solutions do not exist, in the previous case, if T is sufficiently small, as we shall show at the end of the paper. In order to compare theorem 1 with the result in [5], we note that an easy analysis for a radial F shows that, if j/h/l is large enough, then jh = 0 does not imply that (P) is not solvable, differently from what happens in the case n = 1, in which x cannot turn around zero and jh = 0 is therefore an obviously necessary condition for the solvability of (P). In the same way one can also check that the first part of (F,)’ is essential for theorem 1. In fact, if F is radial and if the inward component of VF(x) is arbitrarily close to zero for some

On forced

dynamical

systems

491

values of x tending to zero, then small circular orbits around zero solve the problem with a forcing therm h of small norm and mean value zero. Finally, we point out that (F,) is a rather natural condition in order to have a good compactness at infinity and can be substituted by other conditions to the same aim in order to get easy variants of the theorems. In theorem 3, for instance, one can ask instead of (F,) that vF(x).(llxll -lx) d’rver ges in a suitably sublinear way to +oo as x tends to infinity. In this case, of course, (F4) becomes implicit. We work with variational methods on the Sobolev space E of the T-periodic functions, valued in IR”, with generalized first derivates in L2, by looking for critical points of the functional J defined bv

on the open set A := (x E El V t E [0, T]: x(t) # 0). Actually, we shall work with a truncation of F which will make J defined on the whole of E. The paper is organized as follows. The second section is concerned with some estimates made on the functional and apriori on the possible solutions. In the third one we prove the theorems stated in the introduction and the above remarks about theorem 3, which will be stated precisely. We point out that many papers have been recently written for the case that the singularity is attractive, starting from [3]. We refer to [2] for some bibliography on the subject. 2. A PRIORI

ESTIMATES

Let U be the subset of A defined by U = (x E A] 3 t E [0, T] such that Ix(t)1 = 1

and

v t E (0, T]:

Ix(t)l

2 1)

and let c0 = XiFGJ(x). LEMMA2.1. Assume (F3). Then for any given h E C(S’) one has c,, > --co. Proof.

By (F,) one can find a constant c such that V IYI 2 1.

IFCV)] 5 41 + 1~1)

(2.2)

Given x E U set ,U = (j;[i(t)l’

dt)““,

then, since by Holder inequality T Ii(t)1 dt 5 1 + T”‘p,

x E U * max]x(t)] 5 1 + s0 from (2.2) one has

T

J(x) L ;$

- cT(2 + T”$)

-

Ihl

(s

0

>

(1 + T”Q.

(2.3)

s. SOLMINI

492

Therefore (2.4) LEMMA2.5. Assume that (F,) holds and that liminfF(y)=M>2-$.

(2.6)

y-0

Then we can fix 01> 0 such that if x is a solution to (P) and J(x) 1 co, then $x*r

INI

2 a.

(2.7)

Proof. Assume that, for a given (Y,(2.7) is not true. Then, by using (F2) and by integrating by parts, we get

J(x) I i(c,

=I&

+ cYc,)T + (s

Therefore co+,

+ac,)T+

‘thlo

(s namely

0

0

-

T&f

>

FO). a

>

(2.8)

- 7;;‘kf F(Y) CL

co + T ,f-rfUF(Y) - (c/2)T CYl c27-

+

So%1

(2.9)

’

Since the lim inf of the right hand side of (2.9) is strictly positive as cx -, 0, then (2.9) is false if CYis small enough and therefore the statement of the lemma is true. n We are now in a position to prove the main result in this section. PROPOSITION2.10. Let (F,), (F,) and (2.6) hold and also assume M>Y where y is a constant which will be determined during the proof. Then one can find p > 0 such that if x is a solution to (P) and J(x) I co then

(2.11)

, e$n,

(2.12)

Proof.

I4

2 P.

Let x be a solution to (P) and let to E [0, T] be such that

IxUo)l= t;o”“qI.wl*

(2.13)

Then we have d21x12

,,zUo)

5 0,

namely a(t,).x(t,)

+ lk(fo)12 I 0.

(2.14)

Since x solves (P), this means kto)12 5 (vF(x(to)) as follows from (F,).

+

Wo))

*x(fo)

5

CI

+

(~2

+

Ibk.Jlx(toX

(2.15)

On forced

dynamical

systems

493

Moreover we can always suppose /3 I 1, so we can assume by contradiction Ix(t)] I 1 +

_T Ix(t)/ dt !0

that (2.16)

v t E [O, T].

If we multiply both the sides of the equation in (P) by 2 and we integrate, we get the energy estimate -‘I h(f)z?(t) dt. (2.17) tM)12 = tMto)12 - F&(0) + F(x(to)) .\IO Since the previous lemma tells that we can fix (Y> 0 such that Ix(t,)l L 01, and, like for (2.2), we can deduce from (F,) that for some constant c(a)

v IYI 1 a,

IuY)I 5 C(~)(l+ Irl) we get in this way

(2.18)

lm(to))I5 C(~)U+ Ix(to)lh

(2.19)

Similarly, from (2.6) and (F,) we know that a constant cj exists such that FCV) 2 -c,(l

+

0x(0)

vytro

Irl)

and therefore 2 -c,(l

Now let r = f zoa”-

+

(2.20)

IxWl).

(2.21)

Mol.

From (2.13, (2.16), (2.17), (2.19), (2.21) we deduce tr2 5 *cc, +

(C,

+

Ilm(l + Tc-)+ (CC4 + c,)(2+ 0 + ( hi;. 50

(2.22)

From (2.22) we deduce (2.23)


where c is a constant depending on (c, , c2, c3, CY,llhll, T). By substituting we have F(x(t)) I +2 + c *lhl + c(cY)(l + c) = y. s0 From (2.11) and (2.24), the existence of a constant follows. n

(2.23) in (2.17), (2.24)

p > 0 such that (2.12) holds easily

We close this section by stating a sharper version of lemma 2.1 for the case in which (F4) holds with the constant N finite. Without any restriction, we can assume, after an easy transformation, that in (F4) we have R = 1 and N = 0. Assume that h E C(S’) is given and that h has mean value zero. Then let g E C(S’) be the primitive of h of mean value zero. LEMMA2.25. Assume (F4). Then co > -+jJg(t)]2

dt.

494

S.

SOLIMINI

Proof. By (F4) we have that, if x E U, then constant 6 such that:

t: F(x(t)) I 0 and that we can find a positive

V

vy~R’suchthatl~~~~~l+~:F(y)<-6.

(2.26)

With the same notation of lemma 2.1, since, by HCilder inequality, x(t) has to stay in the annulus B(0, 1 + 6)\B(O, 1) for a time r which satisfies the estimate 6 I #*, we have F(x(t)) dt I -$.

(2.27)

0

Again by Hiilder inequality and by an integration

by parts, we have (2.28)

Therefore,

combining (2.27) and (2.28), we get

co 2 iqf(ip2

- ,u(~~lg~*)“*

+ $)

> inf($’

3. PROOF

OF

- ~(j:~gl*)“*)

THE

= -kj:ig(r)[*

dt.

n

THEOREMS

Proof of theorem 1. Assume by contradiction that we can choose a sequence h, E C(S’) such that tk,ll, -+ 0 T

v~EN:

h,(t)dt

=0

(3.1)

i0 and that we can fix for any n a solution x,, to (P) for h = h,. Let t, be a point of maximum or Ix, (. From (2.14) we get ~n((t,)‘xn(t,)

and therefore,

5 -+#,)I2

IO

since x, solves (P),

(3.2) From (3.2) we obtain

xn(tn)> -Ih,(t,)l. vm, (4 1)*-Ixn(Ml -

(3.3)

From (3.1), (3.3) and (F,)’ we shall deduce that limlx,(t,)l

= +co.

In fact, if x,(l,) -+ x, from the first”part of (F,)‘, we get x # 0, since h,(t,) holds. Then we can pass to the limit in (3.3) and we get in this way

W(X)& 2 1x1

0

(3.4) + 0 and (3.3)

(3.5)

On forced

in contradiction account (F,)‘,

dynamical

systems

495

with the second part of (F,)‘. By an integration by parts, we have, taking into

Therefore the diameter d,, of the orbit x,([O, T]) is bounded, by Holder inequality, by

4, 5 ~ll~nII~21x,(fn)l”2.

(3.6)

By (3.4)-(3.6) one easily gets that

I>

= 0.

(3.7)

Combining (3.7) with (F,)’ we know that if n is large enough one has

v t

E

[O, T]:

VF(x,(t))*x,(t,)

< 0.

(3.8)

Now we multiply both the sides of the equation in (P) by x,(1,,) and we use (3.8) and (3.1). By integrating we have -x,,(,,)~oTx,,(r) dt = joTx,,(t,).VFx,,(t))

dt + x,(t,)

(3.9)

From (3.9) we have T in(f) # 0

I0 in contradiction

to the fact that x, is T-periodic.

n

From now on we shall always assume that h is given and that we are in the situation of theorem 2. For a given K > 0 let pk be a smooth real function such that vx
&z(x) = x 0 I

@k(X) I

1

&(x) = 0 For k > ;n:F(y)

VXEIR

(3.10)

vxzk+l.

the potential Fk is defined by FICY) =

RY)

if IyI 2 1

(~k(Fo1))

if lyl I

(3.11) i

is a C’ potential on RR since one can extend it to zero by taking Fk(0) = pk(k + 1) 5 k + 1. Note that if F satisfies (F2) and (F,) then also Fk does the same and the constants cr and c, are left invariant. Since F = Fk lR”\B(O, 1) then the constant co in lemma

0 =

*aDuanbasqns %U!~J~AUOD E s~q N 3 “(“x) uaql (*@‘[A ;!I

(s)

%‘dl J=

(W”~ $!I

(e)

lvql qDns 3 30 sluamala 30 a>uanbas I! s! N3’(“x) 31 *Jaqtunu It?aJ ua@ lCue s! 3 aJaqM ‘ ‘[‘S’d]

UO!l!pUO3

alEXUS-SiE[Ed

%I!MOI[OJ

aql

SayS!lES

‘f

: y A Uaql

‘0 #

L/,0! la?

‘()Z’E

VUUUUS~

%.~!~ol~oJ aql paau ahi u1p2syl 0~ * qf leuoyun3 aql 01 [g] uraJoaq11u!od aIppes s,zlymou!qE~ Qdde 01 uopisod e u! aJe aM leq1 aas ahi (6I’f)-(fI’Q UJOJ~

[I ‘013 s.I3 k 7 ((S)A)V[ XEuI3U!

-0, = y3:!

(6I’E)

aAeq ahi aJo3aJaqJ *fl 3 (@Auaql 11 5 I(l

(81‘E)

lEq1qsns IL ‘01 3 1 E)lI ‘01 3 43u! = S

ias aM ‘J 3 Aua@ e ~03 ‘31ieql a1oN ’ zx = (1)X ‘ Ix = (0)X ieql q3ns 8 u! [ 1 ‘01 UJOJJAsqwd aql ssels aql aq J 1aT * !x anleA luwsuo~ qi!~ sl!qJo lwlsuo~ aql !xdq Buyouap lI!ls aJe ati aJaqM

~lr? 30

Z‘I = !JO3

(LI’&)

(g 1*E)-(g1'E)uoJd * yg Icq paDt?ldaJd ql!~ I uoyas

OJ > (!X)Y[ leql a3npap LI!sea ah4 aql u! pauyap [ pzuoyun3 aql aq yr ia1 MEN 0

0

(91-f) 1 > zx ‘u& 3

Zx

‘O3 > y .zx - (ZX)dL-- = y .zx - (ZX)QJ_ls 11 lr?yl q3ns luelsuo~ c xi3 ue3 ah4 O f yj! asu!s put! QJtzauqqns SMOJ%d (Ed) 1cq a3u!S ‘I4 O[ + OJ- < (‘X)Ql _?

(SI’E)

1eql q2ns 1 > 1Ix1‘,d 3 *x x!3 pue (lyI{ + oJ-)l_~ ‘,a

‘(q.YA = @-)%A : d 7 1x1

(PI’E)

3

ueql Ja%%q aq y osle 1ay LA saqduq

y

(E 1’0

5

‘ID!qM

(h-)~T:~~g

1Eql AEM E q3llS U! 7 Xi3 UB3 aM OS ‘JazJE[ PUi? Ja%JEl 7 ayE1 aM 3! a%It?qD lou saop d 1eq1 JOOJd aq130 uoyysuo~ aql u10y nap asp s!11 *saydde OI’Z UoysodoJd uaq1 A c y aye] aM *qhoua B!q s! y 3! y uo puadap lou saop (pz’z) u! paugap A luelsuo~ aql4Ieu!3 pue y 30 1uapuadapy os[e s! EdasJno3 30 ‘y uo puadap iou saop (XJ)J iuaisuo~ aq1 aJo3aJaql pue

(ZI’E)

(x)d = N7d

‘,,d

:,?J 7 lil

3

LA

leq1 Zhq OSy aqei I.IWafi -3ooJd aqi uroy sno!Aqo OSIEs! y uo puadap lou saop x) luwsuos aql leql ise3 aqJ *sploq 5-z mural pue J/OS - z/ ‘3 < y x!3 uw ah4 aJo3aJaqL *y uo IHIWlOS‘S

96P

On forced

dynamical

497

systems

Proof. Let pn = max Ix,(t)]. By (F,) we get the existence of a constant c such that

t EIO.7-l

v Y E KY: IWr)l 5 41 + Irlh

(3.21)

By [P.S.], (a) and (3.21) we get, for a constant c’, ;]i,#)12

(3.22)

dt I ~‘(1 + p,).

s Now if pn diverges, from (3.22) and Holder inequality we have min lx,(t)] 1 pn - T”2c”‘2(1

f E [O,ll

+ P$”

+ 00

(3.23)

and (3.23) implies that VF,(x,,) converges uniformly to zero in view of (F,). By [P.S.], (b) we have (3.24) A!,,+ VF,(x,) + h + 0 in H-r and therefore,

by taking the duality product with the function of constant value 1, we get T

Ifn

s0 Therefore

(3.25)

h + 0.

+

s0

for n large we have T jr,

#

0

i0 in contradiction with the fact that x,, is a T-periodic function. Therefore (p,), EN is bounded. By this fact and by (3.22) we deduce that (x,), EN is a bounded sequence in E. Therefore (x,,), EN has a uniformly converging subsequence and, since VFk is uniformly continuous, (VF,(x,J), E,., has also a uniformly converging subsequence. This n fact, in connection with (3.24), easily implies the statement. The second theorem follows directly from the above arguments. Proof of theorem 2. By Rabinowitz’s saddle point theorem [6], [l] (see, for instance [4, theorem 5.21 with dim Y = 1, which can be applied since we have the estimate (3.19) and the [P.S.] condition) we deduce the existence of a critical point x of Jk such that Jk(x) = inf sup J&(s)) y E r s E to,11

2 co

(3.26)

as follows from (3.19). Therefore proposition 2.10 applies and (2.12) holds. By the above discussion, we are assuming that k has been fixed in such a way that (3.14) holds. From (2.12) and (3.14) we see that x is a solution to problem (P). n From the above theorem and from lemma 2.25 we can deduce theorem 3. Proof of theorem 3. Clearly we just have to take care of the case that h has mean value zero, therefore if N < --oo we are in the situation of lemma 2.25. We can produce a Palais-Smale sequence for Jk (namely a sequence which satisfies [P.S.], (a) and [P.S.], (b)) at a level c 1 co or by a direct saddle point argument, which requires a few estimates or simply by perturbing h

498

s. SOLMIN

with a small constant and by applying theorem 2. Therefore we just have to show that the [P.S.], condition holds for c 2 co. We repeat the proof of lemma 3.20 and we stop to (3.25) which does not give now any contradiction. However, from (3.24) we can deduce that &, tends to -g in L2 and, therefore, that, by integrating by parts,

which does produce a contradiction and gives that P,, is bounded. Therefore, we can continue the proof of lemma 3.20, in which the condition that the mean value of h is not zero is not furthermore used, and we get in this way the [P.S.], condition for c 2 co and, subsequently, theorem 3. The case N = +oo is easier since from (3.24) we can deduce that J(x,J -, -00, and so we get the [P.S.], condition for any c E IR, by the same arguments as above. n COROLLARY 3.27. In the situation of theorem 3 if F has only finitely many critical points and all of them are nondegenerate, if h = 0, then To E R exists such that (P) has a nonconstant solution for Tz To. Proof. Since we find a solution by a saddle point argument, by the results in (41 we can avoid all the solutions which have a Morse index different from 1, see [4]. The fact that this is the case for every constant solution when T is large is shown in [2, lemma 121. n

Finally we show that the previous corollary cannot be extended, in general, to cover the case in which T is small. In fact, we consider the following assumption, stronger than (F2) but still compatible with (FJ, 3 k, E > 0 such that: IYI < E * (VF(Y),Y) 5 0 (Irl >

k,

IY - ZI c 4VFCY)l)

(F2)n

=> PF(Y),

Vmz))

> 0.

Note that the second part of (F2)Nis certainly true if the Hessian matrix of F, denoted by V2F is bounded at infinity. If we ask this last assumption the proof of the following proposition becomes simpler. PROPOSITION 3.28. Assume (F,), (F,)” and (F,). Then a constant To > 0 exists such that (P) has only constant solutions if T s To, when h = 0. Proof.

Assume that T is given and that x is a solution to P. If

ma IW)l

OSrST

(3.29)

< E,

then, by an integration by parts, one has from the first part of (F$‘) that ‘/i(t .r0

dt = -

‘i(t)

s0

- x(t) dt =

‘VF(x(t))

* x(t) dt I 0.

(3.30)

s0

Therefore, if (3.29) holds, we can deduce from (3.30) that x is a constant orbit. Hence we may assume that (3.29) is false and therefore we have (2.7) with cx = E. So we are in a position to

On forced

repeat the argument used in proposition stant p > 0. Now let u = J!, and let

dynamical

systems

499

2.10 and we get in this way (2.12) for a suitable con-

L =

sup

VF(x(t)).

OdfST

I L. Since

From (P) we know that v t : Iti

r v=o r0

and

V t, f: Iv(t) - u(T)1 I LT

(3.31)

we get for any given f~ S’ Tlu(i)l = 1TV(T) - j:v(t)l

5 jo’ju(t)

- u(f)1 5 LT’.

(3.32)

Therefore IIull_,I LT. If L = 0 we have that x is a constant orbit and the proof is finished. So assume L # 0 and let f be such that (3.33)

IVF(x(f)) I = L . Now let T2 c E, we claim that Ix(T)1 5 k, where k is as in (F,)“. In fact, if not f ijul

v t : Ix(t) - x(f)1 I

15

I LT2 I cIVF(x(f))l.

(3.34)

I

From (3.34) and (F,)” 7 VF(x(T)) *

T

-VF(x(f))

z(t) =

s0

* VF(x(t))

c 0.

.r0

This contradicts the fact that x is a periodic function and proves the claim. Now we deduce from that, by also using (3.34), that if T c &1’2 vt:psIx(t)lsk+LT’

(3.35)

and L I LySLlpfv-(y)I < -a.

2

(3.36)

If we also ask T < 1, and if we set M=

by differenting

sup Iv2KY>I 3 fiSblSk+L

(3.37)

in (P), we have - ii = V+(x) - u

(3.38)

and therefore I;ti= Since

-~~ri*u=~orV2~(x)(u)~~~MS6$. lu = 0, s0

(3.39)

500

S. SOLIMINI

by expanding u in Fourier series, we see that

From (3.40) and (3.39) we finally deduce that if x is not constant then u # 0 and

which proves the statement. REFERENCES I. AMBROSETTI A. & RABINOWITZ P., Dual variational methods in critical 2. 3. 4. 5. 6.

point theory and applications. J. Func. Analysis 14, 349-381 (1973). COTI ZELATI V., Dynamical systems with effective-like potential. Nonlineur Analysis 12, 209-222 (1988). GORDON W., Conservative dynamical systems involving strong forces. Trans. AMS, 113-135 (1975). LAZER A. & SOLIMINI S., Nontrivial solutions of operator equations and Morse indices of critical points of min-max type. Nonlinear Analysis 12, 761-775 (1988). LAZER A. & SOLIMINI S., On periodic solutions of nonlinear differential equations with singularity. Proc. AMS 2, 109-l 14 (1988). RABINOWUZ P. H., Some minimax theorems and applications to nonlinear partial differential equations. Nonlinear Analysis: a collection of paper in honor of Erich H. Rothe (Edited by L. CESARI, R. KANNAN and H. F. WEINBERGER), pp. 161-177. Academic Press, New York (1978).